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Probability, Fourth Edition by Rick Durrett

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5Martingales

A martingale Xn can be thought of as the fortune at time n of a player who is betting on a fair game; submartingales (supermartingales) as the outcome of betting on a favorable (unfavorable) game. There are two basic facts about martingales. The first is that you cannot make money betting on them (see Theorem 5.2.5), and in particular if you choose to stop playing at some bounded time N, then your expected winnings EXN are equal to your initial fortune X0. (We are supposing for the moment that X0 is not random.) Our second fact, Theorem 5.2.8, concerns submartingales. To use a heuristic we learned from Mike Brennan, “They are the stochastic analogues of nondecreasing sequences and so if they are bounded above (to be precise, sup ...

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